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Some Convexity Results for the Cartan Decomposition

  Published:2003-10-01
 Printed: Oct 2003
  • P. Graczyk
  • P. Sawyer
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Abstract

In this paper, we consider the set $\mathcal{S} = a(e^X K e^Y)$ where $a(g)$ is the abelian part in the Cartan decomposition of $g$. This is exactly the support of the measure intervening in the product formula for the spherical functions on symmetric spaces of noncompact type. We give a simple description of that support in the case of $\SL(3,\mathbf{F})$ where $\mathbf{F} = \mathbf{R}$, $\mathbf{C}$ or $\mathbf{H}$. In particular, we show that $\mathcal{S}$ is convex. We also give an application of our result to the description of singular values of a product of two arbitrary matrices with prescribed singular values.
Keywords: convexity theorems, Cartan decomposition, spherical functions, product formula, semisimple Lie groups, singular values convexity theorems, Cartan decomposition, spherical functions, product formula, semisimple Lie groups, singular values
MSC Classifications: 43A90, 53C35, 15A18 show english descriptions Spherical functions [See also 22E45, 22E46, 33C55]
Symmetric spaces [See also 32M15, 57T15]
Eigenvalues, singular values, and eigenvectors
43A90 - Spherical functions [See also 22E45, 22E46, 33C55]
53C35 - Symmetric spaces [See also 32M15, 57T15]
15A18 - Eigenvalues, singular values, and eigenvectors
 

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